LMFDB - Embedded newform 4338.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4338,2,Mod(1,4338)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4338 = 2 \cdot 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4338.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$34.6391043968$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 20x^{5} + 26x^{4} + 95x^{3} - 121x^{2} - 126x + 158 $ x^7 - x^6 - 20x^5 + 26x^4 + 95x^3 - 121x^2 - 126*x + 158
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1446)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.49848$ of defining polynomial
Character	$\chi$	$=$	4338.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.78201 q^{5} +4.79149 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.78201 q^{5} +4.79149 q^{7} +1.00000 q^{8} -1.78201 q^{10} -5.16992 q^{11} -0.740872 q^{13} +4.79149 q^{14} +1.00000 q^{16} -5.50349 q^{17} +7.53236 q^{19} -1.78201 q^{20} -5.16992 q^{22} -0.852030 q^{23} -1.82442 q^{25} -0.740872 q^{26} +4.79149 q^{28} +7.96141 q^{29} -1.39999 q^{31} +1.00000 q^{32} -5.50349 q^{34} -8.53850 q^{35} +7.50349 q^{37} +7.53236 q^{38} -1.78201 q^{40} +10.5385 q^{41} +0.134906 q^{43} -5.16992 q^{44} -0.852030 q^{46} +9.77995 q^{47} +15.9584 q^{49} -1.82442 q^{50} -0.740872 q^{52} +6.91900 q^{53} +9.21288 q^{55} +4.79149 q^{56} +7.96141 q^{58} +6.17047 q^{59} -2.05145 q^{61} -1.39999 q^{62} +1.00000 q^{64} +1.32024 q^{65} -13.3368 q^{67} -5.50349 q^{68} -8.53850 q^{70} +9.09639 q^{71} -5.44295 q^{73} +7.50349 q^{74} +7.53236 q^{76} -24.7716 q^{77} -0.150051 q^{79} -1.78201 q^{80} +10.5385 q^{82} +8.61873 q^{83} +9.80730 q^{85} +0.134906 q^{86} -5.16992 q^{88} -15.9537 q^{89} -3.54988 q^{91} -0.852030 q^{92} +9.77995 q^{94} -13.4228 q^{95} -11.9349 q^{97} +15.9584 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) $ 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8	$ 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8 - 5 * q^10 - 4 * q^11 + 14 * q^13 + 7 * q^14 + 7 * q^16 - 4 * q^17 + 7 * q^19 - 5 * q^20 - 4 * q^22 + q^23 + 14 * q^25 + 14 * q^26 + 7 * q^28 - 3 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^34 + 17 * q^35 + 18 * q^37 + 7 * q^38 - 5 * q^40 - 3 * q^41 + 11 * q^43 - 4 * q^44 + q^46 + 18 * q^47 + 20 * q^49 + 14 * q^50 + 14 * q^52 + 9 * q^53 - 4 * q^55 + 7 * q^56 - 3 * q^58 - 6 * q^59 + 31 * q^61 + 8 * q^62 + 7 * q^64 + 6 * q^65 + 4 * q^67 - 4 * q^68 + 17 * q^70 + 3 * q^71 + 16 * q^73 + 18 * q^74 + 7 * q^76 + 34 * q^79 - 5 * q^80 - 3 * q^82 - 10 * q^83 + 34 * q^85 + 11 * q^86 - 4 * q^88 - 24 * q^89 + 40 * q^91 + q^92 + 18 * q^94 + q^95 + 27 * q^97 + 20 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.78201	−0.796941	−0.398471	−	0.917181i	$-0.630459\pi$
$5$	−1.78201	−0.796941	−0.398471	+	0.917181i	$0.630459\pi$
$6$	0	0
$7$	4.79149	1.81101	0.905506	−	0.424333i	$-0.139491\pi$
$7$	4.79149	1.81101	0.905506	+	0.424333i	$0.139491\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.78201	−0.563523
$11$	−5.16992	−1.55879	−0.779395	−	0.626533i	$-0.784473\pi$
$11$	−5.16992	−1.55879	−0.779395	+	0.626533i	$0.784473\pi$
$12$	0	0
$13$	−0.740872	−0.205481	−0.102740	−	0.994708i	$-0.532761\pi$
$13$	−0.740872	−0.205481	−0.102740	+	0.994708i	$0.532761\pi$
$14$	4.79149	1.28058
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.50349	−1.33479	−0.667396	−	0.744703i	$-0.732591\pi$
$17$	−5.50349	−1.33479	−0.667396	+	0.744703i	$0.732591\pi$
$18$	0	0
$19$	7.53236	1.72804	0.864021	−	0.503456i	$-0.167938\pi$
$19$	7.53236	1.72804	0.864021	+	0.503456i	$0.167938\pi$
$20$	−1.78201	−0.398471
$21$	0	0
$22$	−5.16992	−1.10223
$23$	−0.852030	−0.177661	−0.0888303	−	0.996047i	$-0.528313\pi$
$23$	−0.852030	−0.177661	−0.0888303	+	0.996047i	$0.528313\pi$
$24$	0	0
$25$	−1.82442	−0.364885
$26$	−0.740872	−0.145297
$27$	0	0
$28$	4.79149	0.905506
$29$	7.96141	1.47840	0.739198	−	0.673488i	$-0.235205\pi$
$29$	7.96141	1.47840	0.739198	+	0.673488i	$0.235205\pi$
$30$	0	0
$31$	−1.39999	−0.251446	−0.125723	−	0.992065i	$-0.540125\pi$
$31$	−1.39999	−0.251446	−0.125723	+	0.992065i	$0.540125\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.50349	−0.943840
$35$	−8.53850	−1.44327
$36$	0	0
$37$	7.50349	1.23357	0.616783	−	0.787133i	$-0.288436\pi$
$37$	7.50349	1.23357	0.616783	+	0.787133i	$0.288436\pi$
$38$	7.53236	1.22191
$39$	0	0
$40$	−1.78201	−0.281761
$41$	10.5385	1.64584	0.822919	−	0.568159i	$-0.192344\pi$
$41$	10.5385	1.64584	0.822919	+	0.568159i	$0.192344\pi$
$42$	0	0
$43$	0.134906	0.0205730	0.0102865	−	0.999947i	$-0.496726\pi$
$43$	0.134906	0.0205730	0.0102865	+	0.999947i	$0.496726\pi$
$44$	−5.16992	−0.779395
$45$	0	0
$46$	−0.852030	−0.125625
$47$	9.77995	1.42655	0.713276	−	0.700883i	$-0.247211\pi$
$47$	9.77995	1.42655	0.713276	+	0.700883i	$0.247211\pi$
$48$	0	0
$49$	15.9584	2.27977
$50$	−1.82442	−0.258012
$51$	0	0
$52$	−0.740872	−0.102740
$53$	6.91900	0.950398	0.475199	−	0.879878i	$-0.342376\pi$
$53$	6.91900	0.950398	0.475199	+	0.879878i	$0.342376\pi$
$54$	0	0
$55$	9.21288	1.24226
$56$	4.79149	0.640290
$57$	0	0
$58$	7.96141	1.04538
$59$	6.17047	0.803326	0.401663	−	0.915787i	$-0.368432\pi$
$59$	6.17047	0.803326	0.401663	+	0.915787i	$0.368432\pi$
$60$	0	0
$61$	−2.05145	−0.262661	−0.131331	−	0.991339i	$-0.541925\pi$
$61$	−2.05145	−0.262661	−0.131331	+	0.991339i	$0.541925\pi$
$62$	−1.39999	−0.177799
$63$	0	0
$64$	1.00000	0.125000
$65$	1.32024	0.163756
$66$	0	0
$67$	−13.3368	−1.62935	−0.814675	−	0.579918i	$-0.803085\pi$
$67$	−13.3368	−1.62935	−0.814675	+	0.579918i	$0.803085\pi$
$68$	−5.50349	−0.667396
$69$	0	0
$70$	−8.53850	−1.02055
$71$	9.09639	1.07954	0.539771	−	0.841812i	$-0.318511\pi$
$71$	9.09639	1.07954	0.539771	+	0.841812i	$0.318511\pi$
$72$	0	0
$73$	−5.44295	−0.637049	−0.318524	−	0.947915i	$-0.603187\pi$
$73$	−5.44295	−0.637049	−0.318524	+	0.947915i	$0.603187\pi$
$74$	7.50349	0.872263
$75$	0	0
$76$	7.53236	0.864021
$77$	−24.7716	−2.82299
$78$	0	0
$79$	−0.150051	−0.0168821	−0.00844103	−	0.999964i	$-0.502687\pi$
$79$	−0.150051	−0.0168821	−0.00844103	+	0.999964i	$0.502687\pi$
$80$	−1.78201	−0.199235
$81$	0	0
$82$	10.5385	1.16378
$83$	8.61873	0.946029	0.473014	−	0.881055i	$-0.343166\pi$
$83$	8.61873	0.946029	0.473014	+	0.881055i	$0.343166\pi$
$84$	0	0
$85$	9.80730	1.06375
$86$	0.134906	0.0145473
$87$	0	0
$88$	−5.16992	−0.551115
$89$	−15.9537	−1.69109	−0.845547	−	0.533901i	$-0.820725\pi$
$89$	−15.9537	−1.69109	−0.845547	+	0.533901i	$0.820725\pi$
$90$	0	0
$91$	−3.54988	−0.372128
$92$	−0.852030	−0.0888303
$93$	0	0
$94$	9.77995	1.00872
$95$	−13.4228	−1.37715
$96$	0	0
$97$	−11.9349	−1.21181	−0.605903	−	0.795538i	$-0.707188\pi$
$97$	−11.9349	−1.21181	−0.605903	+	0.795538i	$0.707188\pi$
$98$	15.9584	1.61204
$99$	0	0
$100$	−1.82442	−0.182442
$101$	−14.3640	−1.42927	−0.714636	−	0.699496i	$-0.753408\pi$
$101$	−14.3640	−1.42927	−0.714636	+	0.699496i	$0.753408\pi$
$102$	0	0
$103$	17.2216	1.69689	0.848446	−	0.529281i	$-0.177538\pi$
$103$	17.2216	1.69689	0.848446	+	0.529281i	$0.177538\pi$
$104$	−0.740872	−0.0726484
$105$	0	0
$106$	6.91900	0.672033
$107$	16.4444	1.58974	0.794872	−	0.606777i	$-0.207538\pi$
$107$	16.4444	1.58974	0.794872	+	0.606777i	$0.207538\pi$
$108$	0	0
$109$	13.3180	1.27563	0.637815	−	0.770190i	$-0.279838\pi$
$109$	13.3180	1.27563	0.637815	+	0.770190i	$0.279838\pi$
$110$	9.21288	0.878413
$111$	0	0
$112$	4.79149	0.452753
$113$	−12.7444	−1.19889	−0.599446	−	0.800415i	$-0.704613\pi$
$113$	−12.7444	−1.19889	−0.599446	+	0.800415i	$0.704613\pi$
$114$	0	0
$115$	1.51833	0.141585
$116$	7.96141	0.739198
$117$	0	0
$118$	6.17047	0.568038
$119$	−26.3699	−2.41733
$120$	0	0
$121$	15.7281	1.42983
$122$	−2.05145	−0.185730
$123$	0	0
$124$	−1.39999	−0.125723
$125$	12.1612	1.08773
$126$	0	0
$127$	1.40717	0.124866	0.0624332	−	0.998049i	$-0.480114\pi$
$127$	1.40717	0.124866	0.0624332	+	0.998049i	$0.480114\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.32024	0.115793
$131$	9.22137	0.805675	0.402838	−	0.915272i	$-0.368024\pi$
$131$	9.22137	0.805675	0.402838	+	0.915272i	$0.368024\pi$
$132$	0	0
$133$	36.0912	3.12951
$134$	−13.3368	−1.15212
$135$	0	0
$136$	−5.50349	−0.471920
$137$	10.0279	0.856742	0.428371	−	0.903603i	$-0.359088\pi$
$137$	10.0279	0.856742	0.428371	+	0.903603i	$0.359088\pi$
$138$	0	0
$139$	−14.4493	−1.22557	−0.612787	−	0.790248i	$-0.709952\pi$
$139$	−14.4493	−1.22557	−0.612787	+	0.790248i	$0.709952\pi$
$140$	−8.53850	−0.721635
$141$	0	0
$142$	9.09639	0.763352
$143$	3.83025	0.320301
$144$	0	0
$145$	−14.1874	−1.17820
$146$	−5.44295	−0.450462
$147$	0	0
$148$	7.50349	0.616783
$149$	−12.7177	−1.04187	−0.520937	−	0.853595i	$-0.674417\pi$
$149$	−12.7177	−1.04187	−0.520937	+	0.853595i	$0.674417\pi$
$150$	0	0
$151$	−1.18244	−0.0962256	−0.0481128	−	0.998842i	$-0.515321\pi$
$151$	−1.18244	−0.0962256	−0.0481128	+	0.998842i	$0.515321\pi$
$152$	7.53236	0.610955
$153$	0	0
$154$	−24.7716	−1.99615
$155$	2.49481	0.200388
$156$	0	0
$157$	−1.20755	−0.0963729	−0.0481864	−	0.998838i	$-0.515344\pi$
$157$	−1.20755	−0.0963729	−0.0481864	+	0.998838i	$0.515344\pi$
$158$	−0.150051	−0.0119374
$159$	0	0
$160$	−1.78201	−0.140881
$161$	−4.08249	−0.321746
$162$	0	0
$163$	20.1989	1.58210	0.791048	−	0.611754i	$-0.209536\pi$
$163$	20.1989	1.58210	0.791048	+	0.611754i	$0.209536\pi$
$164$	10.5385	0.822919
$165$	0	0
$166$	8.61873	0.668943
$167$	15.3193	1.18544	0.592721	−	0.805408i	$-0.298054\pi$
$167$	15.3193	1.18544	0.592721	+	0.805408i	$0.298054\pi$
$168$	0	0
$169$	−12.4511	−0.957778
$170$	9.80730	0.752185
$171$	0	0
$172$	0.134906	0.0102865
$173$	7.35291	0.559031	0.279516	−	0.960141i	$-0.409826\pi$
$173$	7.35291	0.559031	0.279516	+	0.960141i	$0.409826\pi$
$174$	0	0
$175$	−8.74171	−0.660811
$176$	−5.16992	−0.389698
$177$	0	0
$178$	−15.9537	−1.19578
$179$	−20.9444	−1.56546	−0.782728	−	0.622364i	$-0.786173\pi$
$179$	−20.9444	−1.56546	−0.782728	+	0.622364i	$0.786173\pi$
$180$	0	0
$181$	−4.94111	−0.367270	−0.183635	−	0.982995i	$-0.558786\pi$
$181$	−4.94111	−0.367270	−0.183635	+	0.982995i	$0.558786\pi$
$182$	−3.54988	−0.263135
$183$	0	0
$184$	−0.852030	−0.0628125
$185$	−13.3713	−0.983080
$186$	0	0
$187$	28.4526	2.08066
$188$	9.77995	0.713276
$189$	0	0
$190$	−13.4228	−0.973791
$191$	15.2460	1.10316	0.551580	−	0.834122i	$-0.314025\pi$
$191$	15.2460	1.10316	0.551580	+	0.834122i	$0.314025\pi$
$192$	0	0
$193$	−13.9918	−1.00716	−0.503578	−	0.863950i	$-0.667983\pi$
$193$	−13.9918	−1.00716	−0.503578	+	0.863950i	$0.667983\pi$
$194$	−11.9349	−0.856876
$195$	0	0
$196$	15.9584	1.13988
$197$	1.53241	0.109180	0.0545899	−	0.998509i	$-0.482615\pi$
$197$	1.53241	0.109180	0.0545899	+	0.998509i	$0.482615\pi$
$198$	0	0
$199$	−16.4405	−1.16544	−0.582720	−	0.812673i	$-0.698011\pi$
$199$	−16.4405	−1.16544	−0.582720	+	0.812673i	$0.698011\pi$
$200$	−1.82442	−0.129006
$201$	0	0
$202$	−14.3640	−1.01065
$203$	38.1470	2.67740
$204$	0	0
$205$	−18.7798	−1.31164
$206$	17.2216	1.19988
$207$	0	0
$208$	−0.740872	−0.0513702
$209$	−38.9417	−2.69365
$210$	0	0
$211$	10.4483	0.719288	0.359644	−	0.933090i	$-0.382898\pi$
$211$	10.4483	0.719288	0.359644	+	0.933090i	$0.382898\pi$
$212$	6.91900	0.475199
$213$	0	0
$214$	16.4444	1.12412
$215$	−0.240405	−0.0163955
$216$	0	0
$217$	−6.70805	−0.455372
$218$	13.3180	0.902006
$219$	0	0
$220$	9.21288	0.621132
$221$	4.07738	0.274274
$222$	0	0
$223$	19.9124	1.33343	0.666717	−	0.745311i	$-0.267699\pi$
$223$	19.9124	1.33343	0.666717	+	0.745311i	$0.267699\pi$
$224$	4.79149	0.320145
$225$	0	0
$226$	−12.7444	−0.847745
$227$	0.772540	0.0512753	0.0256377	−	0.999671i	$-0.491838\pi$
$227$	0.772540	0.0512753	0.0256377	+	0.999671i	$0.491838\pi$
$228$	0	0
$229$	18.1285	1.19797	0.598983	−	0.800762i	$-0.295572\pi$
$229$	18.1285	1.19797	0.598983	+	0.800762i	$0.295572\pi$
$230$	1.51833	0.100116
$231$	0	0
$232$	7.96141	0.522692
$233$	3.92584	0.257190	0.128595	−	0.991697i	$-0.458953\pi$
$233$	3.92584	0.257190	0.128595	+	0.991697i	$0.458953\pi$
$234$	0	0
$235$	−17.4280	−1.13688
$236$	6.17047	0.401663
$237$	0	0
$238$	−26.3699	−1.70931
$239$	−9.39170	−0.607499	−0.303749	−	0.952752i	$-0.598239\pi$
$239$	−9.39170	−0.607499	−0.303749	+	0.952752i	$0.598239\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	15.7281	1.01104
$243$	0	0
$244$	−2.05145	−0.131331
$245$	−28.4380	−1.81684
$246$	0	0
$247$	−5.58051	−0.355080
$248$	−1.39999	−0.0888996
$249$	0	0
$250$	12.1612	0.769143
$251$	−1.10632	−0.0698304	−0.0349152	−	0.999390i	$-0.511116\pi$
$251$	−1.10632	−0.0698304	−0.0349152	+	0.999390i	$0.511116\pi$
$252$	0	0
$253$	4.40493	0.276935
$254$	1.40717	0.0882938
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.1468	−1.13197	−0.565983	−	0.824417i	$-0.691503\pi$
$257$	−18.1468	−1.13197	−0.565983	+	0.824417i	$0.691503\pi$
$258$	0	0
$259$	35.9529	2.23400
$260$	1.32024	0.0818781
$261$	0	0
$262$	9.22137	0.569698
$263$	16.1400	0.995236	0.497618	−	0.867396i	$-0.334208\pi$
$263$	16.1400	0.995236	0.497618	+	0.867396i	$0.334208\pi$
$264$	0	0
$265$	−12.3298	−0.757411
$266$	36.0912	2.21289
$267$	0	0
$268$	−13.3368	−0.814675
$269$	20.3579	1.24124	0.620622	−	0.784110i	$-0.286880\pi$
$269$	20.3579	1.24124	0.620622	+	0.784110i	$0.286880\pi$
$270$	0	0
$271$	−0.133059	−0.00808276	−0.00404138	−	0.999992i	$-0.501286\pi$
$271$	−0.133059	−0.00808276	−0.00404138	+	0.999992i	$0.501286\pi$
$272$	−5.50349	−0.333698
$273$	0	0
$274$	10.0279	0.605808
$275$	9.43213	0.568779
$276$	0	0
$277$	−2.75086	−0.165283	−0.0826415	−	0.996579i	$-0.526336\pi$
$277$	−2.75086	−0.165283	−0.0826415	+	0.996579i	$0.526336\pi$
$278$	−14.4493	−0.866611
$279$	0	0
$280$	−8.53850	−0.510273
$281$	10.4950	0.626080	0.313040	−	0.949740i	$-0.398653\pi$
$281$	10.4950	0.626080	0.313040	+	0.949740i	$0.398653\pi$
$282$	0	0
$283$	−11.8002	−0.701447	−0.350724	−	0.936479i	$-0.614064\pi$
$283$	−11.8002	−0.701447	−0.350724	+	0.936479i	$0.614064\pi$
$284$	9.09639	0.539771
$285$	0	0
$286$	3.83025	0.226487
$287$	50.4951	2.98063
$288$	0	0
$289$	13.2884	0.781670
$290$	−14.1874	−0.833110
$291$	0	0
$292$	−5.44295	−0.318524
$293$	−2.33984	−0.136695	−0.0683476	−	0.997662i	$-0.521773\pi$
$293$	−2.33984	−0.136695	−0.0683476	+	0.997662i	$0.521773\pi$
$294$	0	0
$295$	−10.9959	−0.640204
$296$	7.50349	0.436131
$297$	0	0
$298$	−12.7177	−0.736717
$299$	0.631245	0.0365058
$300$	0	0
$301$	0.646401	0.0372579
$302$	−1.18244	−0.0680417
$303$	0	0
$304$	7.53236	0.432011
$305$	3.65572	0.209326
$306$	0	0
$307$	−10.7761	−0.615023	−0.307511	−	0.951544i	$-0.599496\pi$
$307$	−10.7761	−0.615023	−0.307511	+	0.951544i	$0.599496\pi$
$308$	−24.7716	−1.41149
$309$	0	0
$310$	2.49481	0.141696
$311$	−6.21016	−0.352146	−0.176073	−	0.984377i	$-0.556339\pi$
$311$	−6.21016	−0.352146	−0.176073	+	0.984377i	$0.556339\pi$
$312$	0	0
$313$	−31.8577	−1.80071	−0.900353	−	0.435161i	$-0.856691\pi$
$313$	−31.8577	−1.80071	−0.900353	+	0.435161i	$0.856691\pi$
$314$	−1.20755	−0.0681459
$315$	0	0
$316$	−0.150051	−0.00844103
$317$	−15.1409	−0.850399	−0.425199	−	0.905100i	$-0.639796\pi$
$317$	−15.1409	−0.850399	−0.425199	+	0.905100i	$0.639796\pi$
$318$	0	0
$319$	−41.1599	−2.30451
$320$	−1.78201	−0.0996177
$321$	0	0
$322$	−4.08249	−0.227508
$323$	−41.4543	−2.30658
$324$	0	0
$325$	1.35166	0.0749768
$326$	20.1989	1.11871
$327$	0	0
$328$	10.5385	0.581892
$329$	46.8605	2.58350
$330$	0	0
$331$	−17.7322	−0.974651	−0.487326	−	0.873220i	$-0.662028\pi$
$331$	−17.7322	−0.974651	−0.487326	+	0.873220i	$0.662028\pi$
$332$	8.61873	0.473014
$333$	0	0
$334$	15.3193	0.838234
$335$	23.7664	1.29850
$336$	0	0
$337$	−18.9845	−1.03415	−0.517076	−	0.855940i	$-0.672979\pi$
$337$	−18.9845	−1.03415	−0.517076	+	0.855940i	$0.672979\pi$
$338$	−12.4511	−0.677251
$339$	0	0
$340$	9.80730	0.531875
$341$	7.23785	0.391952
$342$	0	0
$343$	42.9239	2.31767
$344$	0.134906	0.00727364
$345$	0	0
$346$	7.35291	0.395295
$347$	−24.3826	−1.30892	−0.654462	−	0.756095i	$-0.727105\pi$
$347$	−24.3826	−1.30892	−0.654462	+	0.756095i	$0.727105\pi$
$348$	0	0
$349$	32.2379	1.72566	0.862828	−	0.505498i	$-0.168691\pi$
$349$	32.2379	1.72566	0.862828	+	0.505498i	$0.168691\pi$
$350$	−8.74171	−0.467264
$351$	0	0
$352$	−5.16992	−0.275558
$353$	17.0741	0.908760	0.454380	−	0.890808i	$-0.349861\pi$
$353$	17.0741	0.908760	0.454380	+	0.890808i	$0.349861\pi$
$354$	0	0
$355$	−16.2099	−0.860332
$356$	−15.9537	−0.845547
$357$	0	0
$358$	−20.9444	−1.10694
$359$	14.2429	0.751713	0.375857	−	0.926678i	$-0.377349\pi$
$359$	14.2429	0.751713	0.375857	+	0.926678i	$0.377349\pi$
$360$	0	0
$361$	37.7365	1.98613
$362$	−4.94111	−0.259699
$363$	0	0
$364$	−3.54988	−0.186064
$365$	9.69941	0.507690
$366$	0	0
$367$	−36.3056	−1.89513	−0.947567	−	0.319556i	$-0.896466\pi$
$367$	−36.3056	−1.89513	−0.947567	+	0.319556i	$0.896466\pi$
$368$	−0.852030	−0.0444151
$369$	0	0
$370$	−13.3713	−0.695142
$371$	33.1523	1.72118
$372$	0	0
$373$	6.10048	0.315871	0.157935	−	0.987449i	$-0.449516\pi$
$373$	6.10048	0.315871	0.157935	+	0.987449i	$0.449516\pi$
$374$	28.4526	1.47125
$375$	0	0
$376$	9.77995	0.504362
$377$	−5.89838	−0.303782
$378$	0	0
$379$	0.319854	0.0164298	0.00821491	−	0.999966i	$-0.497385\pi$
$379$	0.319854	0.0164298	0.00821491	+	0.999966i	$0.497385\pi$
$380$	−13.4228	−0.688574
$381$	0	0
$382$	15.2460	0.780052
$383$	−7.78918	−0.398009	−0.199004	−	0.979999i	$-0.563771\pi$
$383$	−7.78918	−0.398009	−0.199004	+	0.979999i	$0.563771\pi$
$384$	0	0
$385$	44.1434	2.24976
$386$	−13.9918	−0.712166
$387$	0	0
$388$	−11.9349	−0.605903
$389$	−21.4359	−1.08684	−0.543422	−	0.839460i	$-0.682872\pi$
$389$	−21.4359	−1.08684	−0.543422	+	0.839460i	$0.682872\pi$
$390$	0	0
$391$	4.68914	0.237140
$392$	15.9584	0.806019
$393$	0	0
$394$	1.53241	0.0772018
$395$	0.267393	0.0134540
$396$	0	0
$397$	35.4141	1.77738	0.888690	−	0.458508i	$-0.151616\pi$
$397$	35.4141	1.77738	0.888690	+	0.458508i	$0.151616\pi$
$398$	−16.4405	−0.824090
$399$	0	0
$400$	−1.82442	−0.0912212
$401$	−21.2685	−1.06210	−0.531049	−	0.847341i	$-0.678202\pi$
$401$	−21.2685	−1.06210	−0.531049	+	0.847341i	$0.678202\pi$
$402$	0	0
$403$	1.03721	0.0516674
$404$	−14.3640	−0.714636
$405$	0	0
$406$	38.1470	1.89320
$407$	−38.7924	−1.92287
$408$	0	0
$409$	28.0691	1.38793	0.693963	−	0.720011i	$-0.255863\pi$
$409$	28.0691	1.38793	0.693963	+	0.720011i	$0.255863\pi$
$410$	−18.7798	−0.927467
$411$	0	0
$412$	17.2216	0.848446
$413$	29.5657	1.45483
$414$	0	0
$415$	−15.3587	−0.753929
$416$	−0.740872	−0.0363242
$417$	0	0
$418$	−38.9417	−1.90470
$419$	−15.4065	−0.752656	−0.376328	−	0.926487i	$-0.622813\pi$
$419$	−15.4065	−0.752656	−0.376328	+	0.926487i	$0.622813\pi$
$420$	0	0
$421$	20.4804	0.998153	0.499077	−	0.866558i	$-0.333673\pi$
$421$	20.4804	0.998153	0.499077	+	0.866558i	$0.333673\pi$
$422$	10.4483	0.508613
$423$	0	0
$424$	6.91900	0.336016
$425$	10.0407	0.487045
$426$	0	0
$427$	−9.82950	−0.475683
$428$	16.4444	0.794872
$429$	0	0
$430$	−0.240405	−0.0115933
$431$	2.10061	0.101183	0.0505914	−	0.998719i	$-0.483889\pi$
$431$	2.10061	0.101183	0.0505914	+	0.998719i	$0.483889\pi$
$432$	0	0
$433$	−0.807534	−0.0388076	−0.0194038	−	0.999812i	$-0.506177\pi$
$433$	−0.807534	−0.0388076	−0.0194038	+	0.999812i	$0.506177\pi$
$434$	−6.70805	−0.321997
$435$	0	0
$436$	13.3180	0.637815
$437$	−6.41780	−0.307005
$438$	0	0
$439$	−1.34347	−0.0641201	−0.0320601	−	0.999486i	$-0.510207\pi$
$439$	−1.34347	−0.0641201	−0.0320601	+	0.999486i	$0.510207\pi$
$440$	9.21288	0.439207
$441$	0	0
$442$	4.07738	0.193941
$443$	15.3870	0.731059	0.365530	−	0.930800i	$-0.380888\pi$
$443$	15.3870	0.731059	0.365530	+	0.930800i	$0.380888\pi$
$444$	0	0
$445$	28.4298	1.34770
$446$	19.9124	0.942881
$447$	0	0
$448$	4.79149	0.226377
$449$	15.5678	0.734690	0.367345	−	0.930085i	$-0.380267\pi$
$449$	15.5678	0.734690	0.367345	+	0.930085i	$0.380267\pi$
$450$	0	0
$451$	−54.4832	−2.56552
$452$	−12.7444	−0.599446
$453$	0	0
$454$	0.772540	0.0362571
$455$	6.32594	0.296564
$456$	0	0
$457$	−23.4673	−1.09776	−0.548878	−	0.835903i	$-0.684945\pi$
$457$	−23.4673	−1.09776	−0.548878	+	0.835903i	$0.684945\pi$
$458$	18.1285	0.847090
$459$	0	0
$460$	1.51833	0.0707925
$461$	9.59182	0.446736	0.223368	−	0.974734i	$-0.428295\pi$
$461$	9.59182	0.446736	0.223368	+	0.974734i	$0.428295\pi$
$462$	0	0
$463$	−18.4306	−0.856544	−0.428272	−	0.903650i	$-0.640877\pi$
$463$	−18.4306	−0.856544	−0.428272	+	0.903650i	$0.640877\pi$
$464$	7.96141	0.369599
$465$	0	0
$466$	3.92584	0.181861
$467$	8.98165	0.415621	0.207811	−	0.978169i	$-0.433366\pi$
$467$	8.98165	0.415621	0.207811	+	0.978169i	$0.433366\pi$
$468$	0	0
$469$	−63.9031	−2.95077
$470$	−17.4280	−0.803894
$471$	0	0
$472$	6.17047	0.284019
$473$	−0.697454	−0.0320689
$474$	0	0
$475$	−13.7422	−0.630536
$476$	−26.3699	−1.20866
$477$	0	0
$478$	−9.39170	−0.429567
$479$	−35.0467	−1.60133	−0.800663	−	0.599114i	$-0.795519\pi$
$479$	−35.0467	−1.60133	−0.800663	+	0.599114i	$0.795519\pi$
$480$	0	0
$481$	−5.55912	−0.253474
$482$	1.00000	0.0455488
$483$	0	0
$484$	15.7281	0.714913
$485$	21.2682	0.965738
$486$	0	0
$487$	3.62045	0.164058	0.0820291	−	0.996630i	$-0.473860\pi$
$487$	3.62045	0.164058	0.0820291	+	0.996630i	$0.473860\pi$
$488$	−2.05145	−0.0928648
$489$	0	0
$490$	−28.4380	−1.28470
$491$	−15.0767	−0.680402	−0.340201	−	0.940353i	$-0.610495\pi$
$491$	−15.0767	−0.680402	−0.340201	+	0.940353i	$0.610495\pi$
$492$	0	0
$493$	−43.8155	−1.97335
$494$	−5.58051	−0.251079
$495$	0	0
$496$	−1.39999	−0.0628615
$497$	43.5853	1.95507
$498$	0	0
$499$	−20.9701	−0.938750	−0.469375	−	0.882999i	$-0.655521\pi$
$499$	−20.9701	−0.938750	−0.469375	+	0.882999i	$0.655521\pi$
$500$	12.1612	0.543866
$501$	0	0
$502$	−1.10632	−0.0493775
$503$	22.0260	0.982090	0.491045	−	0.871134i	$-0.336615\pi$
$503$	22.0260	0.982090	0.491045	+	0.871134i	$0.336615\pi$
$504$	0	0
$505$	25.5969	1.13905
$506$	4.40493	0.195823
$507$	0	0
$508$	1.40717	0.0624332
$509$	−24.1792	−1.07173	−0.535863	−	0.844305i	$-0.680014\pi$
$509$	−24.1792	−1.07173	−0.535863	+	0.844305i	$0.680014\pi$
$510$	0	0
$511$	−26.0798	−1.15370
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.1468	−0.800421
$515$	−30.6891	−1.35232
$516$	0	0
$517$	−50.5616	−2.22370
$518$	35.9529	1.57968
$519$	0	0
$520$	1.32024	0.0578965
$521$	−42.1339	−1.84592	−0.922959	−	0.384898i	$-0.874236\pi$
$521$	−42.1339	−1.84592	−0.922959	+	0.384898i	$0.874236\pi$
$522$	0	0
$523$	37.3346	1.63253	0.816263	−	0.577680i	$-0.196042\pi$
$523$	37.3346	1.63253	0.816263	+	0.577680i	$0.196042\pi$
$524$	9.22137	0.402838
$525$	0	0
$526$	16.1400	0.703738
$527$	7.70484	0.335628
$528$	0	0
$529$	−22.2740	−0.968437
$530$	−12.3298	−0.535571
$531$	0	0
$532$	36.0912	1.56475
$533$	−7.80768	−0.338188
$534$	0	0
$535$	−29.3042	−1.26693
$536$	−13.3368	−0.576062
$537$	0	0
$538$	20.3579	0.877693
$539$	−82.5035	−3.55368
$540$	0	0
$541$	−15.0059	−0.645153	−0.322577	−	0.946543i	$-0.604549\pi$
$541$	−15.0059	−0.645153	−0.322577	+	0.946543i	$0.604549\pi$
$542$	−0.133059	−0.00571537
$543$	0	0
$544$	−5.50349	−0.235960
$545$	−23.7328	−1.01660
$546$	0	0
$547$	−3.60252	−0.154033	−0.0770163	−	0.997030i	$-0.524539\pi$
$547$	−3.60252	−0.154033	−0.0770163	+	0.997030i	$0.524539\pi$
$548$	10.0279	0.428371
$549$	0	0
$550$	9.43213	0.402187
$551$	59.9682	2.55473
$552$	0	0
$553$	−0.718968	−0.0305736
$554$	−2.75086	−0.116873
$555$	0	0
$556$	−14.4493	−0.612787
$557$	−9.30430	−0.394236	−0.197118	−	0.980380i	$-0.563158\pi$
$557$	−9.30430	−0.394236	−0.197118	+	0.980380i	$0.563158\pi$
$558$	0	0
$559$	−0.0999480	−0.00422735
$560$	−8.53850	−0.360818
$561$	0	0
$562$	10.4950	0.442705
$563$	−43.7612	−1.84432	−0.922158	−	0.386813i	$-0.873576\pi$
$563$	−43.7612	−1.84432	−0.922158	+	0.386813i	$0.873576\pi$
$564$	0	0
$565$	22.7107	0.955447
$566$	−11.8002	−0.495998
$567$	0	0
$568$	9.09639	0.381676
$569$	−13.6491	−0.572198	−0.286099	−	0.958200i	$-0.592359\pi$
$569$	−13.6491	−0.572198	−0.286099	+	0.958200i	$0.592359\pi$
$570$	0	0
$571$	31.2605	1.30821	0.654105	−	0.756404i	$-0.273046\pi$
$571$	31.2605	1.30821	0.654105	+	0.756404i	$0.273046\pi$
$572$	3.83025	0.160151
$573$	0	0
$574$	50.4951	2.10763
$575$	1.55446	0.0648256
$576$	0	0
$577$	0.0669611	0.00278763	0.00139381	−	0.999999i	$-0.499556\pi$
$577$	0.0669611	0.00278763	0.00139381	+	0.999999i	$0.499556\pi$
$578$	13.2884	0.552724
$579$	0	0
$580$	−14.1874	−0.589098
$581$	41.2966	1.71327
$582$	0	0
$583$	−35.7707	−1.48147
$584$	−5.44295	−0.225231
$585$	0	0
$586$	−2.33984	−0.0966580
$587$	−3.53480	−0.145897	−0.0729484	−	0.997336i	$-0.523241\pi$
$587$	−3.53480	−0.145897	−0.0729484	+	0.997336i	$0.523241\pi$
$588$	0	0
$589$	−10.5452	−0.434509
$590$	−10.9959	−0.452693
$591$	0	0
$592$	7.50349	0.308391
$593$	−46.5682	−1.91233	−0.956164	−	0.292831i	$-0.905403\pi$
$593$	−46.5682	−1.91233	−0.956164	+	0.292831i	$0.905403\pi$
$594$	0	0
$595$	46.9916	1.92647
$596$	−12.7177	−0.520937
$597$	0	0
$598$	0.631245	0.0258135
$599$	−11.6504	−0.476022	−0.238011	−	0.971263i	$-0.576495\pi$
$599$	−11.6504	−0.476022	−0.238011	+	0.971263i	$0.576495\pi$
$600$	0	0
$601$	−16.7884	−0.684814	−0.342407	−	0.939552i	$-0.611242\pi$
$601$	−16.7884	−0.684814	−0.342407	+	0.939552i	$0.611242\pi$
$602$	0.646401	0.0263453
$603$	0	0
$604$	−1.18244	−0.0481128
$605$	−28.0277	−1.13949
$606$	0	0
$607$	−3.73912	−0.151766	−0.0758831	−	0.997117i	$-0.524178\pi$
$607$	−3.73912	−0.151766	−0.0758831	+	0.997117i	$0.524178\pi$
$608$	7.53236	0.305478
$609$	0	0
$610$	3.65572	0.148016
$611$	−7.24569	−0.293129
$612$	0	0
$613$	−7.58022	−0.306162	−0.153081	−	0.988214i	$-0.548920\pi$
$613$	−7.58022	−0.306162	−0.153081	+	0.988214i	$0.548920\pi$
$614$	−10.7761	−0.434887
$615$	0	0
$616$	−24.7716	−0.998077
$617$	6.90459	0.277968	0.138984	−	0.990295i	$-0.455616\pi$
$617$	6.90459	0.277968	0.138984	+	0.990295i	$0.455616\pi$
$618$	0	0
$619$	30.9729	1.24491	0.622453	−	0.782657i	$-0.286136\pi$
$619$	30.9729	1.24491	0.622453	+	0.782657i	$0.286136\pi$
$620$	2.49481	0.100194
$621$	0	0
$622$	−6.21016	−0.249005
$623$	−76.4422	−3.06259
$624$	0	0
$625$	−12.5494	−0.501974
$626$	−31.8577	−1.27329
$627$	0	0
$628$	−1.20755	−0.0481864
$629$	−41.2954	−1.64655
$630$	0	0
$631$	45.3340	1.80472	0.902359	−	0.430984i	$-0.141834\pi$
$631$	45.3340	1.80472	0.902359	+	0.430984i	$0.141834\pi$
$632$	−0.150051	−0.00596871
$633$	0	0
$634$	−15.1409	−0.601323
$635$	−2.50760	−0.0995111
$636$	0	0
$637$	−11.8231	−0.468448
$638$	−41.1599	−1.62953
$639$	0	0
$640$	−1.78201	−0.0704403
$641$	1.59001	0.0628016	0.0314008	−	0.999507i	$-0.490003\pi$
$641$	1.59001	0.0628016	0.0314008	+	0.999507i	$0.490003\pi$
$642$	0	0
$643$	−46.2269	−1.82301	−0.911505	−	0.411290i	$-0.865078\pi$
$643$	−46.2269	−1.82301	−0.911505	+	0.411290i	$0.865078\pi$
$644$	−4.08249	−0.160873
$645$	0	0
$646$	−41.4543	−1.63100
$647$	−45.1867	−1.77647	−0.888237	−	0.459386i	$-0.848070\pi$
$647$	−45.1867	−1.77647	−0.888237	+	0.459386i	$0.848070\pi$
$648$	0	0
$649$	−31.9008	−1.25222
$650$	1.35166	0.0530166
$651$	0	0
$652$	20.1989	0.791048
$653$	42.1564	1.64971	0.824854	−	0.565345i	$-0.191257\pi$
$653$	42.1564	1.64971	0.824854	+	0.565345i	$0.191257\pi$
$654$	0	0
$655$	−16.4326	−0.642076
$656$	10.5385	0.411459
$657$	0	0
$658$	46.8605	1.82681
$659$	24.0545	0.937030	0.468515	−	0.883456i	$-0.344789\pi$
$659$	24.0545	0.937030	0.468515	+	0.883456i	$0.344789\pi$
$660$	0	0
$661$	8.09972	0.315043	0.157521	−	0.987516i	$-0.449650\pi$
$661$	8.09972	0.315043	0.157521	+	0.987516i	$0.449650\pi$
$662$	−17.7322	−0.689183
$663$	0	0
$664$	8.61873	0.334472
$665$	−64.3151	−2.49403
$666$	0	0
$667$	−6.78336	−0.262653
$668$	15.3193	0.592721
$669$	0	0
$670$	23.7664	0.918175
$671$	10.6058	0.409434
$672$	0	0
$673$	−5.71171	−0.220170	−0.110085	−	0.993922i	$-0.535112\pi$
$673$	−5.71171	−0.220170	−0.110085	+	0.993922i	$0.535112\pi$
$674$	−18.9845	−0.731255
$675$	0	0
$676$	−12.4511	−0.478889
$677$	−4.74578	−0.182395	−0.0911976	−	0.995833i	$-0.529069\pi$
$677$	−4.74578	−0.182395	−0.0911976	+	0.995833i	$0.529069\pi$
$678$	0	0
$679$	−57.1860	−2.19460
$680$	9.80730	0.376093
$681$	0	0
$682$	7.23785	0.277152
$683$	−11.1098	−0.425103	−0.212552	−	0.977150i	$-0.568177\pi$
$683$	−11.1098	−0.425103	−0.212552	+	0.977150i	$0.568177\pi$
$684$	0	0
$685$	−17.8699	−0.682773
$686$	42.9239	1.63884
$687$	0	0
$688$	0.134906	0.00514324
$689$	−5.12609	−0.195289
$690$	0	0
$691$	−20.1461	−0.766393	−0.383197	−	0.923667i	$-0.625177\pi$
$691$	−20.1461	−0.766393	−0.383197	+	0.923667i	$0.625177\pi$
$692$	7.35291	0.279516
$693$	0	0
$694$	−24.3826	−0.925550
$695$	25.7489	0.976710
$696$	0	0
$697$	−57.9985	−2.19685
$698$	32.2379	1.22022
$699$	0	0
$700$	−8.74171	−0.330405
$701$	11.7619	0.444241	0.222121	−	0.975019i	$-0.428702\pi$
$701$	11.7619	0.444241	0.222121	+	0.975019i	$0.428702\pi$
$702$	0	0
$703$	56.5190	2.13165
$704$	−5.16992	−0.194849
$705$	0	0
$706$	17.0741	0.642590
$707$	−68.8250	−2.58843
$708$	0	0
$709$	38.0942	1.43066	0.715329	−	0.698787i	$-0.246277\pi$
$709$	38.0942	1.43066	0.715329	+	0.698787i	$0.246277\pi$
$710$	−16.2099	−0.608347
$711$	0	0
$712$	−15.9537	−0.597892
$713$	1.19284	0.0446720
$714$	0	0
$715$	−6.82556	−0.255261
$716$	−20.9444	−0.782728
$717$	0	0
$718$	14.2429	0.531542
$719$	23.1555	0.863554	0.431777	−	0.901980i	$-0.357887\pi$
$719$	23.1555	0.863554	0.431777	+	0.901980i	$0.357887\pi$
$720$	0	0
$721$	82.5170	3.07309
$722$	37.7365	1.40441
$723$	0	0
$724$	−4.94111	−0.183635
$725$	−14.5250	−0.539444
$726$	0	0
$727$	26.9238	0.998548	0.499274	−	0.866444i	$-0.333600\pi$
$727$	26.9238	0.998548	0.499274	+	0.866444i	$0.333600\pi$
$728$	−3.54988	−0.131567
$729$	0	0
$730$	9.69941	0.358991
$731$	−0.742454	−0.0274606
$732$	0	0
$733$	−17.2023	−0.635380	−0.317690	−	0.948195i	$-0.602907\pi$
$733$	−17.2023	−0.635380	−0.317690	+	0.948195i	$0.602907\pi$
$734$	−36.3056	−1.34006
$735$	0	0
$736$	−0.852030	−0.0314062
$737$	68.9502	2.53981
$738$	0	0
$739$	5.63122	0.207148	0.103574	−	0.994622i	$-0.466972\pi$
$739$	5.63122	0.207148	0.103574	+	0.994622i	$0.466972\pi$
$740$	−13.3713	−0.491540
$741$	0	0
$742$	33.1523	1.21706
$743$	18.2940	0.671142	0.335571	−	0.942015i	$-0.391071\pi$
$743$	18.2940	0.671142	0.335571	+	0.942015i	$0.391071\pi$
$744$	0	0
$745$	22.6631	0.830313
$746$	6.10048	0.223354
$747$	0	0
$748$	28.4526	1.04033
$749$	78.7934	2.87905
$750$	0	0
$751$	−20.6843	−0.754781	−0.377391	−	0.926054i	$-0.623179\pi$
$751$	−20.6843	−0.754781	−0.377391	+	0.926054i	$0.623179\pi$
$752$	9.77995	0.356638
$753$	0	0
$754$	−5.89838	−0.214806
$755$	2.10712	0.0766861
$756$	0	0
$757$	47.0362	1.70956	0.854781	−	0.518989i	$-0.173692\pi$
$757$	47.0362	1.70956	0.854781	+	0.518989i	$0.173692\pi$
$758$	0.319854	0.0116176
$759$	0	0
$760$	−13.4228	−0.486895
$761$	−11.1450	−0.404005	−0.202002	−	0.979385i	$-0.564745\pi$
$761$	−11.1450	−0.404005	−0.202002	+	0.979385i	$0.564745\pi$
$762$	0	0
$763$	63.8129	2.31018
$764$	15.2460	0.551580
$765$	0	0
$766$	−7.78918	−0.281435
$767$	−4.57152	−0.165068
$768$	0	0
$769$	−4.34239	−0.156590	−0.0782952	−	0.996930i	$-0.524948\pi$
$769$	−4.34239	−0.156590	−0.0782952	+	0.996930i	$0.524948\pi$
$770$	44.1434	1.59082
$771$	0	0
$772$	−13.9918	−0.503578
$773$	−34.1077	−1.22677	−0.613385	−	0.789784i	$-0.710192\pi$
$773$	−34.1077	−1.22677	−0.613385	+	0.789784i	$0.710192\pi$
$774$	0	0
$775$	2.55418	0.0917488
$776$	−11.9349	−0.428438
$777$	0	0
$778$	−21.4359	−0.768515
$779$	79.3798	2.84408
$780$	0	0
$781$	−47.0276	−1.68278
$782$	4.68914	0.167683
$783$	0	0
$784$	15.9584	0.569942
$785$	2.15187	0.0768035
$786$	0	0
$787$	−30.0320	−1.07053	−0.535263	−	0.844685i	$-0.679788\pi$
$787$	−30.0320	−1.07053	−0.535263	+	0.844685i	$0.679788\pi$
$788$	1.53241	0.0545899
$789$	0	0
$790$	0.267393	0.00951342
$791$	−61.0647	−2.17121
$792$	0	0
$793$	1.51986	0.0539719
$794$	35.4141	1.25680
$795$	0	0
$796$	−16.4405	−0.582720
$797$	−10.6696	−0.377936	−0.188968	−	0.981983i	$-0.560514\pi$
$797$	−10.6696	−0.377936	−0.188968	+	0.981983i	$0.560514\pi$
$798$	0	0
$799$	−53.8238	−1.90415
$800$	−1.82442	−0.0645031
$801$	0	0
$802$	−21.2685	−0.751016
$803$	28.1396	0.993025
$804$	0	0
$805$	7.27506	0.256412
$806$	1.03721	0.0365343
$807$	0	0
$808$	−14.3640	−0.505324
$809$	17.8924	0.629062	0.314531	−	0.949247i	$-0.398153\pi$
$809$	17.8924	0.629062	0.314531	+	0.949247i	$0.398153\pi$
$810$	0	0
$811$	31.7290	1.11416	0.557078	−	0.830460i	$-0.311922\pi$
$811$	31.7290	1.11416	0.557078	+	0.830460i	$0.311922\pi$
$812$	38.1470	1.33870
$813$	0	0
$814$	−38.7924	−1.35967
$815$	−35.9946	−1.26084
$816$	0	0
$817$	1.01616	0.0355510
$818$	28.0691	0.981411
$819$	0	0
$820$	−18.7798	−0.655818
$821$	1.03778	0.0362188	0.0181094	−	0.999836i	$-0.494235\pi$
$821$	1.03778	0.0362188	0.0181094	+	0.999836i	$0.494235\pi$
$822$	0	0
$823$	−14.1080	−0.491774	−0.245887	−	0.969299i	$-0.579079\pi$
$823$	−14.1080	−0.491774	−0.245887	+	0.969299i	$0.579079\pi$
$824$	17.2216	0.599942
$825$	0	0
$826$	29.5657	1.02872
$827$	−23.4202	−0.814399	−0.407199	−	0.913339i	$-0.633495\pi$
$827$	−23.4202	−0.814399	−0.407199	+	0.913339i	$0.633495\pi$
$828$	0	0
$829$	16.5375	0.574371	0.287185	−	0.957875i	$-0.407280\pi$
$829$	16.5375	0.574371	0.287185	+	0.957875i	$0.407280\pi$
$830$	−15.3587	−0.533108
$831$	0	0
$832$	−0.740872	−0.0256851
$833$	−87.8267	−3.04301
$834$	0	0
$835$	−27.2992	−0.944727
$836$	−38.9417	−1.34683
$837$	0	0
$838$	−15.4065	−0.532208
$839$	−1.24274	−0.0429043	−0.0214521	−	0.999770i	$-0.506829\pi$
$839$	−1.24274	−0.0429043	−0.0214521	+	0.999770i	$0.506829\pi$
$840$	0	0
$841$	34.3841	1.18566
$842$	20.4804	0.705801
$843$	0	0
$844$	10.4483	0.359644
$845$	22.1881	0.763292
$846$	0	0
$847$	75.3610	2.58943
$848$	6.91900	0.237599
$849$	0	0
$850$	10.0407	0.344393
$851$	−6.39320	−0.219156
$852$	0	0
$853$	21.2009	0.725905	0.362952	−	0.931808i	$-0.381769\pi$
$853$	21.2009	0.725905	0.362952	+	0.931808i	$0.381769\pi$
$854$	−9.82950	−0.336359
$855$	0	0
$856$	16.4444	0.562059
$857$	40.1206	1.37049	0.685247	−	0.728311i	$-0.259694\pi$
$857$	40.1206	1.37049	0.685247	+	0.728311i	$0.259694\pi$
$858$	0	0
$859$	9.41186	0.321129	0.160564	−	0.987025i	$-0.448669\pi$
$859$	9.41186	0.321129	0.160564	+	0.987025i	$0.448669\pi$
$860$	−0.240405	−0.00819773
$861$	0	0
$862$	2.10061	0.0715470
$863$	11.9175	0.405677	0.202838	−	0.979212i	$-0.434983\pi$
$863$	11.9175	0.405677	0.202838	+	0.979212i	$0.434983\pi$
$864$	0	0
$865$	−13.1030	−0.445515
$866$	−0.807534	−0.0274411
$867$	0	0
$868$	−6.70805	−0.227686
$869$	0.775752	0.0263156
$870$	0	0
$871$	9.88086	0.334800
$872$	13.3180	0.451003
$873$	0	0
$874$	−6.41780	−0.217085
$875$	58.2704	1.96990
$876$	0	0
$877$	9.39083	0.317106	0.158553	−	0.987350i	$-0.449317\pi$
$877$	9.39083	0.317106	0.158553	+	0.987350i	$0.449317\pi$
$878$	−1.34347	−0.0453398
$879$	0	0
$880$	9.21288	0.310566
$881$	21.0903	0.710549	0.355274	−	0.934762i	$-0.384387\pi$
$881$	21.0903	0.710549	0.355274	+	0.934762i	$0.384387\pi$
$882$	0	0
$883$	−14.1010	−0.474538	−0.237269	−	0.971444i	$-0.576252\pi$
$883$	−14.1010	−0.474538	−0.237269	+	0.971444i	$0.576252\pi$
$884$	4.07738	0.137137
$885$	0	0
$886$	15.3870	0.516937
$887$	−1.87876	−0.0630825	−0.0315412	−	0.999502i	$-0.510042\pi$
$887$	−1.87876	−0.0630825	−0.0315412	+	0.999502i	$0.510042\pi$
$888$	0	0
$889$	6.74245	0.226134
$890$	28.4298	0.952970
$891$	0	0
$892$	19.9124	0.666717
$893$	73.6661	2.46514
$894$	0	0
$895$	37.3232	1.24758
$896$	4.79149	0.160072
$897$	0	0
$898$	15.5678	0.519504
$899$	−11.1459	−0.371737
$900$	0	0
$901$	−38.0786	−1.26858
$902$	−54.4832	−1.81409
$903$	0	0
$904$	−12.7444	−0.423873
$905$	8.80513	0.292692
$906$	0	0
$907$	3.43819	0.114163	0.0570816	−	0.998370i	$-0.481820\pi$
$907$	3.43819	0.114163	0.0570816	+	0.998370i	$0.481820\pi$
$908$	0.772540	0.0256377
$909$	0	0
$910$	6.32594	0.209703
$911$	−43.2566	−1.43316	−0.716578	−	0.697507i	$-0.754292\pi$
$911$	−43.2566	−1.43316	−0.716578	+	0.697507i	$0.754292\pi$
$912$	0	0
$913$	−44.5582	−1.47466
$914$	−23.4673	−0.776230
$915$	0	0
$916$	18.1285	0.598983
$917$	44.1841	1.45909
$918$	0	0
$919$	−54.0177	−1.78188	−0.890940	−	0.454122i	$-0.849953\pi$
$919$	−54.0177	−1.78188	−0.890940	+	0.454122i	$0.849953\pi$
$920$	1.51833	0.0500579
$921$	0	0
$922$	9.59182	0.315890
$923$	−6.73926	−0.221825
$924$	0	0
$925$	−13.6895	−0.450109
$926$	−18.4306	−0.605668
$927$	0	0
$928$	7.96141	0.261346
$929$	−32.6046	−1.06972	−0.534861	−	0.844940i	$-0.679636\pi$
$929$	−32.6046	−1.06972	−0.534861	+	0.844940i	$0.679636\pi$
$930$	0	0
$931$	120.204	3.93953
$932$	3.92584	0.128595
$933$	0	0
$934$	8.98165	0.293889
$935$	−50.7030	−1.65816
$936$	0	0
$937$	−32.4067	−1.05868	−0.529340	−	0.848410i	$-0.677560\pi$
$937$	−32.4067	−1.05868	−0.529340	+	0.848410i	$0.677560\pi$
$938$	−63.9031	−2.08651
$939$	0	0
$940$	−17.4280	−0.568439
$941$	−22.1886	−0.723329	−0.361665	−	0.932308i	$-0.617791\pi$
$941$	−22.1886	−0.723329	−0.361665	+	0.932308i	$0.617791\pi$
$942$	0	0
$943$	−8.97912	−0.292400
$944$	6.17047	0.200832
$945$	0	0
$946$	−0.697454	−0.0226762
$947$	30.7941	1.00068	0.500338	−	0.865830i	$-0.333209\pi$
$947$	30.7941	1.00068	0.500338	+	0.865830i	$0.333209\pi$
$948$	0	0
$949$	4.03253	0.130901
$950$	−13.7422	−0.445856
$951$	0	0
$952$	−26.3699	−0.854654
$953$	−12.5993	−0.408130	−0.204065	−	0.978957i	$-0.565415\pi$
$953$	−12.5993	−0.408130	−0.204065	+	0.978957i	$0.565415\pi$
$954$	0	0
$955$	−27.1686	−0.879154
$956$	−9.39170	−0.303749
$957$	0	0
$958$	−35.0467	−1.13231
$959$	48.0486	1.55157
$960$	0	0
$961$	−29.0400	−0.936775
$962$	−5.55912	−0.179233
$963$	0	0
$964$	1.00000	0.0322078
$965$	24.9337	0.802644
$966$	0	0
$967$	−11.5738	−0.372190	−0.186095	−	0.982532i	$-0.559583\pi$
$967$	−11.5738	−0.372190	−0.186095	+	0.982532i	$0.559583\pi$
$968$	15.7281	0.505520
$969$	0	0
$970$	21.2682	0.682880
$971$	−44.0315	−1.41304	−0.706519	−	0.707694i	$-0.749735\pi$
$971$	−44.0315	−1.41304	−0.706519	+	0.707694i	$0.749735\pi$
$972$	0	0
$973$	−69.2336	−2.21953
$974$	3.62045	0.116007
$975$	0	0
$976$	−2.05145	−0.0656653
$977$	21.2797	0.680798	0.340399	−	0.940281i	$-0.389438\pi$
$977$	21.2797	0.680798	0.340399	+	0.940281i	$0.389438\pi$
$978$	0	0
$979$	82.4796	2.63606
$980$	−28.4380	−0.908420
$981$	0	0
$982$	−15.0767	−0.481117
$983$	45.1430	1.43984	0.719919	−	0.694058i	$-0.244179\pi$
$983$	45.1430	1.43984	0.719919	+	0.694058i	$0.244179\pi$
$984$	0	0
$985$	−2.73078	−0.0870099
$986$	−43.8155	−1.39537
$987$	0	0
$988$	−5.58051	−0.177540
$989$	−0.114944	−0.00365501
$990$	0	0
$991$	9.38797	0.298219	0.149109	−	0.988821i	$-0.452359\pi$
$991$	9.38797	0.298219	0.149109	+	0.988821i	$0.452359\pi$
$992$	−1.39999	−0.0444498
$993$	0	0
$994$	43.5853	1.38244
$995$	29.2973	0.928787
$996$	0	0
$997$	−4.42959	−0.140286	−0.0701432	−	0.997537i	$-0.522346\pi$
$997$	−4.42959	−0.140286	−0.0701432	+	0.997537i	$0.522346\pi$
$998$	−20.9701	−0.663796
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4338.2.a.w.1.3		7
3.2	odd	2		1446.2.a.n.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1446.2.a.n.1.5	✓	7	3.2	odd	2
4338.2.a.w.1.3		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4338 = 2 \cdot 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4338.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.78201 q^{5} +4.79149 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.78201 q^{5} +4.79149 q^{7} +1.00000 q^{8} -1.78201 q^{10} -5.16992 q^{11} -0.740872 q^{13} +4.79149 q^{14} +1.00000 q^{16} -5.50349 q^{17} +7.53236 q^{19} -1.78201 q^{20} -5.16992 q^{22} -0.852030 q^{23} -1.82442 q^{25} -0.740872 q^{26} +4.79149 q^{28} +7.96141 q^{29} -1.39999 q^{31} +1.00000 q^{32} -5.50349 q^{34} -8.53850 q^{35} +7.50349 q^{37} +7.53236 q^{38} -1.78201 q^{40} +10.5385 q^{41} +0.134906 q^{43} -5.16992 q^{44} -0.852030 q^{46} +9.77995 q^{47} +15.9584 q^{49} -1.82442 q^{50} -0.740872 q^{52} +6.91900 q^{53} +9.21288 q^{55} +4.79149 q^{56} +7.96141 q^{58} +6.17047 q^{59} -2.05145 q^{61} -1.39999 q^{62} +1.00000 q^{64} +1.32024 q^{65} -13.3368 q^{67} -5.50349 q^{68} -8.53850 q^{70} +9.09639 q^{71} -5.44295 q^{73} +7.50349 q^{74} +7.53236 q^{76} -24.7716 q^{77} -0.150051 q^{79} -1.78201 q^{80} +10.5385 q^{82} +8.61873 q^{83} +9.80730 q^{85} +0.134906 q^{86} -5.16992 q^{88} -15.9537 q^{89} -3.54988 q^{91} -0.852030 q^{92} +9.77995 q^{94} -13.4228 q^{95} -11.9349 q^{97} +15.9584 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) \) 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8	\( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8 - 5 * q^10 - 4 * q^11 + 14 * q^13 + 7 * q^14 + 7 * q^16 - 4 * q^17 + 7 * q^19 - 5 * q^20 - 4 * q^22 + q^23 + 14 * q^25 + 14 * q^26 + 7 * q^28 - 3 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^34 + 17 * q^35 + 18 * q^37 + 7 * q^38 - 5 * q^40 - 3 * q^41 + 11 * q^43 - 4 * q^44 + q^46 + 18 * q^47 + 20 * q^49 + 14 * q^50 + 14 * q^52 + 9 * q^53 - 4 * q^55 + 7 * q^56 - 3 * q^58 - 6 * q^59 + 31 * q^61 + 8 * q^62 + 7 * q^64 + 6 * q^65 + 4 * q^67 - 4 * q^68 + 17 * q^70 + 3 * q^71 + 16 * q^73 + 18 * q^74 + 7 * q^76 + 34 * q^79 - 5 * q^80 - 3 * q^82 - 10 * q^83 + 34 * q^85 + 11 * q^86 - 4 * q^88 - 24 * q^89 + 40 * q^91 + q^92 + 18 * q^94 + q^95 + 27 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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